Hindawi Publishing Corporation
International Journal of Stochastic Analysis
Volume 2011, Article ID 942478, 19 pages
doi:10.1155/2011/942478
Research Article
Optimal Harvesting When the Exchange Rate Is
a Semimartingale
E. R. Offen and E. M. Lungu
Department of Mathematics, Faculty of Science, University of Botswana, Private Bag UB 0022,
Gaborone, Botswana
Correspondence should be addressed to E. R. Offen, elias.offen@gmail.com
Received 8 August 2011; Revised 31 October 2011; Accepted 1 November 2011
Academic Editor: Onesimo Hernandez Lerma
Copyright q 2011 E. R. Offen and E. M. Lungu. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We consider harvesting in the Black-Scholes Quanto Market when the exchange rate is being
modeled by the process Et E0 exp{Xt }, where Xt is a semimartingale, and we ask the following
question: What harvesting strategy γ ∗ and the value function Φ maximize the expected total
income of an investment? We formulate a singular stochastic control problem and give sufficient
conditions for the existence of an optimal strategy. We found that, if the value function is not too
sensitive to changes in the prices of the investments, the problem reduces to that of Lungu and
Øksendal. However, the general solution of this problem still remains elusive.
1. Introduction
This paper is concerned with an optimal harvesting strategy in the Black-Scholes Quanto
Market when the exchange rate is being driven by a general semimartingale. Specifically,
it is proposed that the optimal harvesting strategy can be found under certain conditions.
The paper aims to make a contribution by deriving the general formula for an optimal
harvesting strategy when the exchange rate is a semimartingale. This study could shed light
on the application of general semimartingales in optimization of harvests from investments.
Optimal harvesting is one of the crucial areas in finance because investment into stocks
and bonds can be used as a source of revenue to expand business. Therefore, making
investors happy through an optimal harvesting strategy could lead to more investments and
consequently to further expansion of business. This study will make reference to dividend
policy to illustrate an optimal harvesting strategy. Indeed a suboptimal dividend policy can
result in destruction of shareholder confidence. How much to payout and still maintain
growth of investments has been a challenge. For example, Miller and Modigilliani 1 claimed
that a dividend policy was irrelevant in perfect markets because it had no impact on firm
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International Journal of Stochastic Analysis
value. However, research on dividend policy that followed Miller and Modigilliani 1 has
further examined various market imperfections and have identified the relevance of dividend
policy. A number of stochastic models for optimal dividend policy can also be found in Taksar
2 and the references therein.
A number of these models 1, 3, etc. have developed an optimal harvesting strategy
as optimal stochastic control problems, and that is our approach in this paper. For example,
Asmussen and Taksar 4 applied the theory of singular control in their study of a company’s
optimal dividend policy that tries to maximize expected value of the total discounted
payments to the shareholders. Asmussen and Taksar 4 made an assumption that no fixed
costs are incurred during payment of dividends and that the liquid assets are modeled by
Brownian Motion with drift. Others who also contributed to this problem are Højgaard and
Taksar 5, 6, Radner and Shepp 7, Asmussen et al. 3, and Choulli et al. 8. The studies
5–7 treated the problem as a classical singular stochastic control problem but allowed a
control to affect both potential profits and the risks of the financial corporation. JeanblancPicqué and Shiryaev 9 investigated the problem of a company that tries to maximize
the expected total discounted amount of dividend payments by modeling dividends as
a stochastic impulse control problem. They 9 looked at a situation whereby the company
faced a fixed cost each time a dividend was paid out by choosing optimally the timing and the
size of the payments. They 9 assumed that, when there is no intervention, the liquid asset
follows a Brownian Motion process with drift. Lungu and Øksendal 10 have considered the
problem of optimizing flow of dividends for a market situation with two investments and
determined an optimal harvesting strategy. They concluded that the optimal strategy was to
do nothing as long as the investments were in the nonintervention region but to harvest when
the investment reached a certain calculated value see Lungu and Øksendal 10. Motivated
by Lungu and Øksendal 10, this study considers a similar problem but now in the BlackScholes Quanto market with the exchange rate modeled by a general semimartingale. The
paper is structured as follows. Section 2 states the model and the necessary theory. Section 3
applies the theory, while Section 4 gives the conclusions.
2. The Model
In the absence of interventions, the dynamics of St, the value of the sterling risky
investment, can be modeled by the equation
St S0 exp{αt σWt },
2.1
where α ∈ R is the riskless interest rate, the constant σ > 0 is the volatility, and Wt is Brownian
Motion. Let the sterling to dollar exchange be modeled by the equation
Et E0 exp{Xt },
2.2
and suppose that these processes are on a filtered probability space Ω, F, F, P, where F FS ∪ FE and FS {FtS , t ≥ 0}, FtS σ Su : 0 ≤ u ≤ t is the natural filtration generated
by the stock price process while FE {FtE , t ≥ 0}, FtE σ Eu : 0 ≤ u ≤ t is the natural
filtration generated by the exchange rate process. F describes information about prices and
the exchange rate revealed to investors. We assume that the probability space Ω, F, F, P
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satisfies the usual conditions, that is, the σ-field F is P-complete and every Ft contains all
P-null sets of F. Xt ∈ SemF, P, that is, Xt is a càdlàg process that admits the decomposition
Xt X0 At Mt , t ≥ 0, where At At ∈ V a process of bounded variation, A0 0, and
M Mt ∈ Mloc a local martingale, M0 0.
We consider harvesting from the investment process given 2.1 previously, and we
ask the following question: what value function Φs, y and harvesting strategy γ ∗ t maximise
the total expected discounted utility harvested from a given time interval.
Since our asset is in sterling but our currency of businesses is the dollar, we need first
of all to find the dollar equivalence of this asset. To do this, we let Yt be the dollar value of the
sterling asset price given by
Yt Et · St .
2.3
Using 2.1 and 2.2, we obtain
Yt E0 S0 exp{αt σWt Xt }
Y0 exp{Ht },
2.4
where
Ht αt σWt Xt .
2.5
Ht is a semimartingale since it is the sum of two semimartingales μt
σWt and Xt . This in turn
implies that Yt is a semimartingale. Our approach will be probabilistic rather than statistical;
hence, it becomes reasonable to express Yt in stochastic exponential form.
Theorem 2.1 Ito’s theorem for semimartingales 11. Let Ht be a semimartingale, and let f be
a C2 real function. Then, fHt is again a semimartingale and
fHt fH0 t
0
f Hs− dHs 1
2
t
0
f Hs− dH c s fHs −fHs− −f Hs− ΔHs ,
0≤s≤t
2.6
where ΔHs Hs − Hs− and H c t is the quadratic characteristic of the continuous martingale part
Htc of Ht , that is, a predictable process such that H c 2t − H c t ∈ Mloc .
For the proof of Theorem 2.1, the reader is referred to Protter 11.
In this study, we use Theorem 2.1 to rewrite Yt in stochastic exponential form.
Let
Yt fHt Y0 expHt ,
2.7
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then, using Theorem 2.1, we have
Y t Y0 e
H0
Y0 eH0
1 t Hs− c
Hs
Hs−
Hs−
e
e
dHs dH s − e
ΔHs
e −e
2 0
0
0≤s≤t
2.8
t
1 t Hs−
Hs−
c
Hs
Hs−
Hs−
e −e
e dHs e dH s − e ΔHs
2 0
0
0≤s≤t
t Hs−
and, in differential form, this can be expressed as
dYt Y0 eHt− dHt Y0 eHt− dHt 1 Ht−
e dH c t eHt − eHt− − eHt− ΔHt
2
1 Ht−
c
Ht− ΔHt
Ht−
Ht−
e dH t e
− e − e ΔHt
2
1
eHt− dHt eHt− dH c t eHt− eΔHt − 1 − ΔHt
2
1
Ht−
c
ΔHs
e
− 1 − ΔHs
e d Ht H t 2
0<s≤t
2.9
t ,
Yt− dH
where
t Ht 1 H c t eΔHs − 1 − ΔHs .
H
2
0<s≤t
2.10
We associate with this semimartingale Yt an integer-valued random measure defined as
μn A; ω IA ΔXn ω,
A ∈ B Rd ,
2.11
where I is an indicator function, that is,
μn A; ω 1 if ΔXn ω ∈ A,
0 if ΔXn ω ∈
/ A,
2.12
see Shiryaev 12, for details. We define the integral-valued random measures of jumps
·n≥1 , where μX
A; ω nk1 μX
A, ω and μX
A; ω IΔXk ω ∈ A, A ∈
μX μX
k
k
0,n
0,n
d
BR \ {0} and by ν νω, ds, dx the compensator of μ, that is, the predictable measure
νn A : ω PΔXn ∈ A | Fn−1 ω P-a.s. with the property that μ − ν is a local martingale
measure. This means that, for each A ∈ BR \ {0}, μw : 0, t × A − νω; 0, t × At>0 is a
local martingale with value 0 for t 0.
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Let ϕ be a truncation function, for example, ϕx xI|x| ≤ 1. Then ΔXs −
0 ⇔ |ΔXs | > α for some α > 0. We now denote the jump part of X corresponding to
ϕΔXs /
big jumps by
X̌ ΔXk − ϕΔXk .
0<k≤t
2.13
Using random measure of jumps, 2.9 can be written in canonical form as
dYt Yt− dHt
1
Y− αdt dC ϕ t dH c σdWt dXtc dex − 1 μ − ν
2
d ex − 1 − ϕx μ ,
2.14
where Cϕ is a predictable process and X c is the continuous martingale part of Xt 13.
Adopting the definition of harvesting strategy from Lungu and Øksendal 10, we
define a harvesting strategy as a stochastic process with the following properties: γt, ω ∈
R, t > s, ω ∈ Ω,
1 γt is measurable with respect to σ-algebra Ft generated by Xs, ·; s ≤ t i.e.,
{γt}γ≥0 is adapted;
2 γt, ω is nondecreasing with respect to t for almost all a.a. ω;
3 γt, ω is right continuous as a function of t for a.a. ω;
4 γs, ω 0 for a.a. ω.
γt, ω represents the total amount harvested from an initial time s up to time t. We let Γ
represent a set of all harvesting strategies. If we apply the harvesting strategy γt, ω, then
the corresponding process Y γ satisfies the equation
1
γ
γ
dYt Y− αdt dC ϕ t dH c t σdWt dXtc dex − 1 μ − ν
2
x
d e − 1 − φx μ − dγt,
Y γ s− y1 .
2.15
It is important to note that the difference between Y γ s and Y γ s− : Y γ s− is the state
before harvesting starts at time t s while Y γ s is the state immediately after. If γ consists of
an immediate harvest of size γ at t s, then
Y s Y s− − Δγ.
2.16
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Let the prices/utilities per unit investment when harvested at time t be given by a constant
nonnegative function π. The expected total discounted payoff in this case is given by
J s, y E
γ
y
T
πe
−ρs
t
· dγt ,
2.17
0
where Es,y denotes expectation with respect to probability law Qs,y of Y s,y t t, Y γ t for
t ≥ s, assuming that Y s,y 0 s, y,
T inf{t > 0; πY t ∈
/ S},
S s, y : πe−ρs
t y ≥ 0
2.18
are the time of bankruptcy and S is the solvency region, respectively. The optimal harvesting
problem is then to find the value function Φs, y and an optimal dividend strategy γ ∗ t such
that
∗
Φ s, y supJ γ s, y J γ s, y .
γ∈Γ
2.19
We let 0 < t1 < t2 · · · denote the jumping times of the given strategy γ ∈ Γ, and we let
Δtk γtk − γt−k be the jump of γtk . We also let
γ c t γt −
Δγtk s≤tk ≤t
2.20
be the continuous part of γt. We formulate the sufficient conditions for the given function
φs, y to be the value function Φs, y of 2.19 and for a given strategy γ ∈ Γ to be optimal
in the following theorem.
Theorem 2.2 extended Lungu and Øksendal 10. Suppose that φ ≥ 0 is twice continuously
differentiable on S with the following properties:
1 One has
∂φ
≥ πe−ρs
t .
∂y
2.21
2 One has
1 2
∂φ
∂φ 1 2 ∂φ2
αt C ϕ t σ H c t ey − 1 μ
σ
Lφ t, y ∂t
2
∂y 2 ∂y2
≤0
on S.
2.22
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Then
φ s, y ≥ Φ s, y
2.23
on S.
3 Define the nonintervention region as
D
∂φ t, y ∈ S;
t, y > πe−ρs
t ∀i 1 · · · n .
∂y
2.24
Suppose
Lφ 0
2.25
in D,
and there exists a harvesting strategy γ such that the following hold:
Y γ t ∈ D
∂φ ∂y
∀t > s,
t, Y γ t − πe−ρs
t γ c t 0
∀i 1 · · · n,
i.e., γ c increases only when
2.26
∂φ
πe−ρs
t ,
∂y
where D is the closure of D, that is, D D ∪ ∂D, where ∂D is the boundary of D.
4 One has
∞
1
y
∂φ γ t, Y s αdt dC ϕ t dH c e − 1 − ϕ y dμ
∂y
2
−∞
∞ 1 2
c
ey − 1 − ϕ y dμ dt ≤ 0.
− αt C ϕ t σ H t 2
−∞
2.27
5 One has
γ tk Δφtk : φ tk , Y γ tk − φ tk , Y γ t−k −π · Δ
2.28
at all jumping times tk ≥ s of γ tk and
Es,y φ TR , Y γ TR −→ ∞,
2.29
TR T ∧ R ∧ inf t > s; Y γ t ≥ R .
2.30
where
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Then
φ s, y Φ s, y
∀ s, y ∈ S
2.31
and γ γ ∗ is an optimal harvest strategy.
Proof. Consider the following:
ΔYt αΔt ΔC ϕ t ΔH c t σΔWt ΔXtc Δex − 1 μ − ν Δ ex − 1 − ϕ ν
Δex − 1 μ − ν Δ ex − 1 − ϕ ν,
2.32
1
dYt αdt dC ϕ t dH c σdWt dXtc dey − 1 μ − ν
2
d ex − 1 − ϕx μ − dγt
∞
1
x
c
c
αdt dC ϕ t dH σdWt dXt e −1−ϕ y d μ−ν
2
−∞
∞
x
e − 1 − ϕ y dμ − dγt.
2.33
−∞
Choose γ ∈ Γ, and assume that φ ∈ C2 satisfies 2.21-2.22. Then, by Ito’s formula for
semimartingales and then computing the expectation throughout the equation
Esy φTR , Y γ TR E φs, Y γ s Es,y
s,y
T R
T
R
s
∂φ
t, Y γ sdt
∂t
1
∂φ
s,y
γ
E
t, Y s αdt dC ϕ t dH c t σdWt dXtc
∂y
2
s
∞
∞
x
e − 1 − ϕ y dμ − dγt
ex − 1d μ − ν −∞
−∞
T
2
R
φ
∂
1
σ 2 t, Y γ t 2 t, Y γ Es,y
∂y
s
2
∂φ γ
s,y
γ
γ −
φ
−
E
tk , Y tk ΔYi tk .
φtk , Y tk − φ tk , Y ttk
∂y
s<tk ≤TR
2.34
From the theory of martingales for integrals,
E
s,y
T
R
s
T
T ∞
R
R
∂φ
∂φ
s,y
c
s,y
y
σ dWt E
σ dXt E
e − 1d μ − ν 0.
∂y
∂y
s
s
−∞
2.35
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Substituting 2.28, 2.32, and 2.35 in 2.34, we have
Esy φTR , Y γ TR E φ s, y Es,y
sy
T
R
s
E
s,y
T R
∞
1
x
∂φ
γ
c
e − 1 −ϕx dμ −dγt
t, Y s αdt dC φ t dH t ∂y
2
−∞
∂2 φ
1 2
γ
γ
σ t, X t 2 t, Y 2
∂y
s
Es,y
∂φ
t, Y γ sdt
∂t
T
R
s
E
s,y
s<tk ≤TR
∂φ tk , Y φ tk Δex −1 μ − ν Δ ex −1−ϕ ν
Δφtk , Y tk −
∂y
γ
.
2.36
Now from 2.22,
1 2
∂φ
∂φ
Lφt, x − αt C ϕ t σ H c t ex − 1 − ϕ y μ
∂t
2
∂y
∂2 φ
1
− σ 2 t, x 2 .
2
∂y
2.37
Using 2.37 in 2.36 gives
Esy φTR , Y γ Esy φTR , Y γ T
R
E
s,y
Lφt, xdt
s
E
s,y
T R
s
−E
s,y
T
R
s
E
sx
∞
1
y
∂φ
γ
c
e − 1 − ϕ y dμ
t, Y s αdt dC ϕ t dH ∂y
2
−∞
∞ 1 2
σ H c t − αt C ϕ t ey − 1 − ϕ y dμ dt
2
−∞
∂φ
t, Y γ sdγt
∂y
s<tk ≤TR
x
∂φ φ
x
tk , Y tk Δe − 1 μ − ν Δ e −1−ϕ ν
Δφtk , Y tk −
∂y
γ
2.38
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Using the fact that Esy φTR , Y γ ≤ φs, x and that Lφs, x ≤ 0 refer to 2.27 yields the
inequality
Esy φTR , Y γ ≤ φ s, y − E
s,y
E
sx
T
s
s<tk ≤TR
R
∂φ
γ
t, Y sdγt
∂y
∂φ tk , Y φ tk Δex −1 μ−ν Δ ex −1−φ ν
Δφtk , Y tk −
∂y
γ
.
2.39
From the relation
γ c t γt −
Δγtk ,
s≤tk ≤t
2.40
we deduce that
dγt dγ c t Δt,
ΔY t −Δγt.
2.41
Using 2.41, the right-hand side of 2.32 becomes
Δex − 1 μ − ν Δ ex − 1 − φ ν −Δγt.
2.42
Furthermore, using 2.42 we can achieve the simplification of 2.34 as follows: taking the
last three terms in 2.39 and using the fact that
s≤tk ≤TR
ftk s<tk ≤TR
ftk Δfs,
2.43
we have
x
∂φ φ
x
E
tk , Y tk Δe − 1 μ − ν Δ e − 1 − φ ν
Δφtk , Y tk −
∂y
s<tk ≤TR
x
∂φ sy
γ
φ
x
E
tk , Y tk Δe −1 μ−ν Δ e −1−φ ν
Δφtk , Y tk −
∂y
s≤tk ≤TR
x
s,y
γ
s,y ∂φ
φ
x
s, Y s Δe − 1 μ − ν Δ e − 1 − φ ν
− E Δφs, Y s − E
∂y
∂φ
sy
γ
γ
Δφtk , Y tk −
E
tk , Y tk Δγt
∂y
s≤tk ≤TR
∂φ
− Es,y Δφs, Y γ s Es,y
s, Y γ sΔγs .
∂y
2.44
sy
γ
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Substituting 2.44 into 2.39, inequality 2.39 becomes
∂φ
∂φ
γ
s,y
γ
t, Y sdγt − E
t, Y sΔγt
∂y
s
∂y
s≤tk ≤TR
∂φ sy
γ
sy
φ
E
tk , Y tk Δγt
Δφtk , Y tk E
∂y
s≤tk ≤TR
s≤<tk ≤TR
s,y
γ
s,y ∂φ
γ
− E Δφs, Y s E
s, Y sΔγs .
∂y
2.45
E φTR , Y γ ≤ φ s, y − Es,y
sy
T
R
This will then lead to the inequality
E φTR , Y γ TR ≤ φ s, y − Es,y
sy
E
sy
T
R
s
∂φ
t, Y γ sdγ c t
∂y
2.46
Δφtk , Y tk .
γ
s≤tk ≤TR
By the Mean-Value property, we have
∂φ γ
γ
Δφ tk , Y γ tk tk , Yk ΔYi tk ∂y
2.47
for some point Yk on the line connecting the points Y γ t−k and Y γ tk , and, using 2.41, we
have
γ
∂φ ∂φ γ γ
γ
tk , Yk ΔYi tk −
tk , Xk Δγ tk
Δφ tk , yγ tk ∂y
∂y
2.48
which leads to
E φTR , Y γ TR ≤ φ s, y − Es,y
T
∂φ
t, Y γ sdγ c t
s
∂y
∂φ γ
sy
−E
tk , Xk Δγ tk .
∂y
s≤tk ≤TR
sy
R
2.49
From 2.21, we obtain
φ s, y −
TR
s
∂φ
t, Y γ sdγ c t ≤ φs, x −
∂y
TR
s
π · e−ρ
st dγ c t.
2.50
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International Journal of Stochastic Analysis
Equation 2.21 can also be expressed in discrete form as
∂φ γ Δγ tk ≥
tk , X
πe−ρ
st Δγ tk
k
∂y
s≤tk ≤TR
s≤tk ≤TR
2.51
Combining 2.50 and 2.51 gives the inequality
TR
∂φ ∂φ
γ
tk , Yk Δγ tk
t, Y γ sdγ c t −
∂y
s
∂y
s≤tk ≤TR
TR
≤ φ s, y −
π · e−ρ
st dγ c t −
πe−ρ
st Δγ tk .
φ s, y −
2.52
s≤tk ≤TR
s
Taking the expectation of 2.52 yields the inequality
∂φ
γ
c
t, Y sdγ t
s
∂y
∂φ γ
sy
−E
tk , Xk Δγ tk
∂y
s≤tk ≤TR
T
R
sy
−ρ
st
c
s,y
−ρ
st
≤ φ s, y − E
π ·e
dγ t − E
πe
Δγ tk ,
T
R
E φTR , Y γ TR ≤ φ s, y − Es,y
s,y
s≤tk ≤TR
s
2.53
from which we obtain
φ s, y ≥ E
s,y
T
R
s
≥ Esy
T
R
π ·e
−ρ
st
dγ t E
c
sy
πe
−ρ
st
Δγ tk
Esy φTR , Y γ TR s≤tk ≤TR
π · e−ρ
st dγt Esy φTR , Y γ TR .
s
2.54
Since R < ∞, γ ∈ Γ were arbitrary and φ ≥ 0, this proves that
φ s, y ≥ Φ s, y and γ is optimal.
2.55
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13
Let us now assume that D is given by 2.24 and that 2.25–2.28 hold. If we replace γ in the
above calculation by γ , then equality holds everywhere and we end up with the relation
φ s, y E
s,y
T
R
π ·e
−ρ
st
d
γ Eφ TR , Y TR .
γ
2.56
s
Letting R → ∞ and using 2.29, we get
φ s, y E
s,y
∞
π · e−ρ
st d
γ .
2.57
s
Combining eqntawina with 2.23, we have
φ s, y Φ s, y ,
γ is optimal.
2.58
The strategy γ can be found by solving the Skorohod stochastic differential equations see
Lungu and Øksendal 10.
3. Application of the Theory
From the form of our discounted utility function 2.17, it becomes reasonable to look for the
function Φ of the form
Φ s, y e−ρs Ψ y .
3.1
φ s, y e−ρs ψ y .
3.2
Let
Equations 2.21, 2.22 become
∂ψ
≥ π,
∂y
3.3
1 2
∂ψ
Lψ t, y −ρψ αt C ϕ t σ H c t ex − 1 μ
2
∂y
1 ∂2 ψ
σ2 2 ≤ 0
2 ∂y
3.4
on S,
respectively. We try a solution ψ of the form
ψ y Fz,
where z πy.
3.5
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International Journal of Stochastic Analysis
Equations 3.2 and 3.3 become
πF z ≥ π
F z ≥ 1,
or
1 2
σ H c t ex − 1 μ πF z
AFz −ρFz αt C ϕ t 2
1 2 2 σ π F z
2
−ρFz κF z βF z < 0,
3.6
3.7
where
κ
1 2
σ H c t ex − 1 μ π,
αt C ϕ t 2
1
β σ2π 2.
2
3.8
Inequality 3.7 is similar to inequality 3.11 in Lungu and Øksendal 10. The major
difference is that in Lungu and Øksendal 10 the coefficients are constants whereas in our
case the coefficient κ is not a constant. Since κt, z is a process with jumps, the general
solution of
βF z κF z − ρFz 0
3.9
is elusive. We still try to explore the behavior of the solution after fixing the value of t; that is,
we want to see the general behavior of the solution with respect to z.
3.1. Examples
3.1.1. Example 1 (κ Constant with respect to z)
The case κ is a constant that reduces to the problem in Lungu and Øksendal 10. The
auxiliary equation for 3.9 is
3.10
βm2 κm − ρ 0.
The solutions are
m1,2 −κ ±
κ2 − 4ρβ
2β
.
3.11
Let us suppose the nonintervention region B to be of the form
B {z; 0 < z < z∗ }
3.12
International Journal of Stochastic Analysis
15
for some z∗ > 0. From 2.21 and 2.22, we try a solution of the form
Fz z G,
α1 em1 z α2 em2 z ,
for z ≥ z∗ ,
0 < z < z∗ ,
3.13
for G ∈ R. We want to determine parameters α1 , α2 , G, and z∗ such that F becomes a C2 at
z z∗ . Using the continuity and differentiability of Fz at z z∗ and taking α α1 −α2 ,
we obtain
z∗ 2 lnm2 /m1 > 0,
m1 − m2 α m1 em1 z∗ m2 em2 z∗ −1 ,
3.14
G αem1 z∗ − em2 z∗ − z∗ .
With this choice of parameters, all the conditions of Theorem 2.2 are satisfied, and we have
Φ s, y α1 e−ρs em1 z em2 z ,
e−ρs z G,
for 0 ≤ z < z∗ ,
z∗ ≤ z.
3.15
Similarly, as discussed in Lungu and Øksendal 10, the optimal strategy is obtained by doing
nothing as long as Y t ∈ 0, z∗ i.e., Y t ∈ B and to harvest a total amount γ ∗ γ of the
reflected process Y γ in the direction of −π.
3.1.2. Example 2 (When κ Is Not a Constant)
We now look at a more general case corresponding to κ not a constant. Suppose that we can
write Fz as
1 κ
dz ,
Fz V z exp −
2 β
3.16
then 3.9 can be transformed into its canonical form given by
V z ηzV z 0,
3.17
where
η−
and the jumps are embedded in ηz.
ρ 1 κ 2 1 κ −
−
β 4 β
4 β z
3.18
16
International Journal of Stochastic Analysis
To solve 3.17, we use the jump transfer matrix method 14. The auxiliary equation
of 3.17 is
3.19
m2 ηz 0
with solutions
m1 imz,
m2 −imz,
where mz ηz.
3.20
The general solution for V z can be written as
V z h1 z expimz h2 z exp−imz,
3.21
where h1 z and h2 z are functions to be determined. The solution 3.21 can be expressed
as
V z expΦzt Hz,
3.22
where
Hz h1 z
,
h2 z
Φz imz
−imz
3.23
and the superscript t denotes transpose of a matrix. F is the solution of the equation
dHz UzHzdz,
3.24
U −zK z − exp−zKz Dz−1 CzK z expzKz ,
3.25
where
p−2 ,
p − 1 mq
n×n
Kz mp zδpq n×n ,
Cz p−1
Dx mzq z
,
n×n
K x mp zδpq
3.26
n×n
cf. 14.
Clearly, for the case corresponding to n 2, the matrices 3.26 are given by
0 0
,
1 1
m1 0
K
,
0 m2
C
1 1
,
m1 m 2
m1 0
hence K .
0 m2
D
3.27
International Journal of Stochastic Analysis
17
Using power series expansion for exponential square matrices and truncating the
expansion at second-order terms, we have
zm1 0
expzK exp
0 zm2
1 0
zm1 0
O2 terms
0 1
0 zk2
1 zm1
0
O2 terms.
0
1 zm2
3.28
Similarly,
exp−zK 0
1 − zm1
.
0
1 − zm2
3.29
From 3.25 and 3.28, we obtain an approximation for Uz as
⎤
m2
1
− z
exp−zm1 − m2 ⎥
m1
⎢
m1 − m2
m2 − m1
⎥
⎢
Uz ⎢
⎥.
⎦
⎣ m1
1
exp
zm1 − m2 m2
− z
m1 − m2
m2 − m1
⎡
3.30
From 3.19 and 3.20, we obtain
m z −1 i2mzz expi2zmz
Vz .
2mz exp−2izmz −1 − i2mzz
3.31
The jump transfer matrix from region 1 to region 2 across the interface z ς in our case takes
the form
Q1 → 2
⎡m m
⎤
1
2 iςm2 −m1 m2 − m1 iςm1 m2 e
e
⎢ 2m2
⎥
2m2
⎥
⎢
⎣ m2 − m1 −iςm m m2 m1 −iςm −m ⎦.
1
2
2
1
e
e
2m2
2m2
3.32
18
International Journal of Stochastic Analysis
This jump transfer matrix over singularities given by Qς−δz → ς
δz is simplified as in 14 to
⎡
Qς−δz → ς
δz
Qς−δz → ς
δz
Qς−δz → ς
δz
1
i
⎢ 2
⎢
⎣1 − i
2
⎡
1−i
⎢
⎣ 1 2
i
2
1 0
0 1
⎤
1−i
2 ⎥
⎥
1 i⎦
2
⎤
1
i
2 ⎥
1 − i⎦
2
type A,
3.33
type B,
type C.
The reader is referred to 14 and the references therein for details regarding classification of
singularities.
3.1.3. Analysis of the Results
Let z0 0 and our solutions 3.21 have jumps at points z1 , z2 , z3 , . . . , zn , and we define
Wj z : zj−1 < z < zj
for i 1, 2, 3, . . . , n,
Bj z : zj−1 < z < z∗j
3.34
for some z∗i , and we define
B
n
&
3.35
Bj .
j1
For z ∈ Wi , we define
Fj z for z ≥ zj∗ ,
z Gj ,
hj1 ze
mj1 z
hj2 e
mj2 z
,
F {F0 , F1 , F2 , . . . , Fn },
hj ze−ρs emj1 z emj2 z ,
Φs, zj e−ρs z Gj ,
0 < z < zj∗ ,
for 0 ≤ z < zj∗ ,
3.36
zj∗ ≤ z,
Φs, z {Φ1 s, z, Φ2 s, z, . . . , Φn s, z}.
3.1.4. Conjecture
The optimal strategy is achieved by doing nothing during jumps and as long as Y t ∈ Bj but
to harvest according to local time γj∗ γ at the boundary ∂Bj .
Remarks. In each nonintervention region Bj , Y γ is a reflected process at ∂Bj , since, as
Y hits the boundary, a certain amount is harvested thereby forcing the process to go below
zj∗ .
γ
International Journal of Stochastic Analysis
19
4. Conclusion
Most of the conditions in Theorem 2.2 are extensions from Lungu and Øksendal 10 with
exception of condition 2.5, that is,
∞
1
y
∂φ
γ
c
e − 1 − φ y dμ
t, Y s αdt dC φ t dH ∂y
2
−∞
∞ 1 2
c y
− αt C φ t σ H t e − 1 − φ y dμ dt ≤ 0.
2
−∞
4.1
We note that this condition is achieved if ∂φ/∂y is small. It is found under additional
condition 2.27 that our problem can be reduced to a second-order differential equation
similar to that in Lungu and Øksendal 10 though with some jumps. What is observed is
that if the investment process is being modeled by a semimartingale in general, optimal value
function Φs, y and the optimal dividend strategy can be found if the rate of change of the
value function φs, y with respect to the investment process itself, that is, ∂φ/∂y, is small
enough. In other words, φs, y should not be too sensitive to variations in investments. Our
results further show that the general solution to this problem is still elusive.
References
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